One of the approaches to feasible quantum computation is to look for physical materials that have anyonic topological phases. In this talk I will describe a kind of converse construction: Every unitary spherical category yields a lattice system with an anyonic topological phase via the Turaev-Viro invariant from the field of quantum topological invariants.
Some spherical categories are better than others for quantum computation, and the Turaev-Viro model is not necessarily the most economical or likely Hamiltonian, but it provides an across-the-board starting point. If the spherical category is modular, then the Turaev-Viro model splits into two topological phases with cancelling central charge.
The ideas that I will discuss mostly comes from an old, unpublished discussion with Alexei Kitaev.